Computational approach to contact fatigue damage initiation and deformation analysis of gear teeth fl anks

IJRET: International Journal of Research in Engineering and Technology ISSN: 2319-1163
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Volume: 01 Issue: 04 | Dec-2012, Available @ http://www.ijret.org 564
COMPUTATIONAL APPROACH TO CONTACT FATIGUE DAMAGE
INITIATION AND DEFORMATION ANALYSIS OF GEAR TEETH
FlANKS
Amisha Y. Pathak
Assistant Professor, Mechanical Department, Bhavshiji Polytechnic, Gujarat, India, makodia_amisha@yahoo.co.in
Abstract
The paper describes a general computational model for the simulation of contact fatigue-damage initiation and deformation in the
contact area of meshing gears. The model considers the continuum mechanics approach, where the use of homogenous and elastic
material is assumed. The stress field in the contact area and the relationship between the cyclic contact loading conditions and
observed contact points on the tooth flank are simulated with moving Hertzian contact pressure in the framework of the finite element
method analysis. An equivalent model of Hertzian contact between two cylinders is used for evaluating contact conditions at the major
point of contact of meshing gears. For the purpose of fatigue-damage analysis, the model, which is used for prediction of the number
of loading cycles required for initial fatigue damage to appear, is based on the Coffin-Manson relationship between deformations and
loading cycles. On the basis of computational results, and with consideration of some particular geometrical and material
parameters, the initiation life of contacting spur gears in regard to contact fatigue damage can be estimated.
Index Terms: Contact fatigue, Deformation, Crack initiation, Numerical modeling and Gear teeth flanks
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1. INTRODUCTION
The fatigue process of mechanical elements is a material
characteristic and depends upon cyclic plasticity, local
deformation, dislocation motion and formation of micro and
macro-cracks and their propagation. Contact fatigue is
extremely important for all engineering applications involving
localized contacts such as gears, brakes, clutches, rolling
bearings, wheels, rails, screws and riveted joints. The repeated
rolling and/or sliding contact conditions cause permanent
damage to the material due to accumulation of deformation.
Contact fatigue process can be divided into two main parts: (1)
initiation of micro-cracks due to local accumulation of
dislocations, high stresses in local points, plastic deformation
around inhomogeneous inclusions or other imperfections on or
under contact surface;(2) crack propagation, which causes
permanent damage to a mechanical element, i.e. exceeding the
fracture toughness of the material. The main aim of this paper
is to model contact fatigue crack initiation. Although modeling
of contact fatigue initiation is often supported by experimental
investigations, most of the work published on contact fatigue
is theoretical [1, 2, 5, 12]. The work of Mura and Nakasone
[21] and Cheng et al. [5] represent a large step forward in the
field of developing physical and mathematical models
concerning fatigue-damage initiation. Practical applicability of
those models in engineering is, however, limited. Large
numbers of different material sand geometric parameters are
needed to determine the calculation number of loading cycles
for fatigue-damage initiation [5]. Additionally, those
parameters differ for different materials, geometries and
loading spectra [5]. The applicability of analytical methods is
limited to idealized engineering problems and they are based
upon well-known theoretical methods for determining fatigue-
damage initiation strain-life methods including Coffin-
Manson’s hypothesis, Morrow’s analysis, Smith-Watson-
Topper (SWT) method, etc. [2, 4, 9, 11, 23, 24].
The significance of dealing with different mechanical
elements and constructions and different approaches to sizing
and endurance control proves that those problems are of great
interest nowadays [2, 3, 7, 9, 10, 23]. For designing machines
and devices, dimensioning with respect to service life is
increasingly taken into account. This also applies to gearing
which is still today one of the very important components of
almost all machines. Two kinds of teeth damage can occur on
gears under repeated loading due to fatigue: namely the pitting
of gear teeth flanks and tooth breakage in the tooth root.
Various studies have analyzed numerical approaches to
bending fatigue life in tooth root [2, 13], as well as pitting
resistance of gear teeth flanks [10, 12, 14–17, 22]. The effect
of different surface treatments on the fatigue behavior at the
tooth root of spur gears has also been investigated [3]
In our previous research work, the main focus was on the
bending and pitting analysis of spur gears [2, 10, 12–17] in
regards to fatigue crack growth. The initiation phase of the
material fatigue process is now included. This paper describes
a computational model for contact fatigue crack initiation in

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the contact area of gear teeth flanks. In the presented paper,
the focus is on the numerical determination of damage
initiation at gear teeth flanks. The most common methods of
gear design are based on conventional standard procedures
like DIN [6], AGMA [1] and ISO. Although the standards for
calculation are the most up-to date methods available, they do
not give detailed information of the fatigue life of gears.
However, all standard models are, in some manner, rough and
do not give enough accurate results because they do not take
into account the actual conditions. Therefore, the development
of model sand procedures for calculations of fatigue-damage
initiation of spur gears seems to be a good choice. For this
purpose, a general computational model of contact fatigue
initiation has been developed [24]. Computational models to
fatigue initiation analyses are based upon the method which
determines the ratio between the specific deformation and the
number of loading cycles, often referred to as local stress-
strain method [11, 18, 25]. Lately, a connection between the
results, obtained by means of the finite element method (FEM)
and/or the boundary element method (BEM), and those,
obtained by the analyses of material fatigue [4, 9, 24], can be
traced. Due to the complexity of contact fatigue initiation at
gear teeth flanks, the generalized model of contact fatigue
initiation, presented in [24] is appropriately modified for this
particular application. The loading cycle during meshing of a
gear pair needs to be determined first. The Hertzian boundary
conditions are evaluated at characteristic points along the
engagement line by means of an equivalent model of two
cylinders. The changing characteristics of gear teeth sliding
along the engagement line have been considered by means of
a corresponding coefficient of friction. The resulting stress
loading cycle of meshing gears is then used for the contact
fatigue analysis. The influence of the type and quality of the
contact surface treatment has also been considered—in
computational simulation this is usually done by means of a
corresponding fatigue limit parameter.
The process of surface pitting can be visualized as the
formation of small surface initial cracks which grow under
repeated contact loading. Eventually, the crack becomes large
enough for unstable growth to occur, which causes the
material surface layer to break away. The resulting void is
called the surface pit (Fig. 1).
The number of stress cycles, N, required for the pitting of a
gear teeth flank to occur can be determined from the number
of stress cycles, Ni, required for the appearance of the initial
crack in the material and the number of stress cycles, Np,
required for a crack to propagate from the initial to the critical
crack length, when the final failure can be expected to occur:
N = Ni + Np (1)
However, the prime aim of the present study is to present a
computational model for prediction of contact fatigue
initiation—e.g. Ni in Eq. (1)—which is based on continuum
mechanics, cyclic contact loading and characteristic material
fatigue parameters. The material model is assumed as being
homogeneous, without the imperfections such as inclusions,
asperities, roughness, residual stresses, etc., which often occur
in mechanical elements. Moving contact load is used for
simulation of the cyclic loading in fatigue crack initiation
analysis of the meshing of gears.
Fig -1: Typical surface micro pits on gear tooth flanks
2. DESCRIPTION OF THE PROBLEM AND
COMPUTATIONAL MODEL
2.1Contact fatigue process at meshing gears
Initial surface cracks leading to the pitting of gears are
generally observed to appear in the contact areas where high
normal contact pressure is combined with significant sliding
velocities, which result in additional frictional loading of the
surface material layer. There are several locations where
pitting is apt to occur. However, the most critical contact
loading conditions for initial crack formation and propagation
are identified as the rolling contact with sliding, where the
contact sliding, and with that the effect of friction, is opposite
to the direction of the rolling contact motion [8, 17]. Thus, the
worst contact loading conditions appear when the gear teeth
are in contact at the inner point of single teeth pair
engagement (point B, Fig. 4), where the surface-breaking
initial cracks are expected to develop first. Of course, this fact
is valid for the pinion gear in the case of revolutions reduction
(reduction gear). Furthermore, pinions are apt to pit rather than
gears for two reasons [8]. First, the pinion is ordinarily the
driver (reduction gear). The directions of sliding are such that
sliding is away from the pitch line on the driver and toward
the pitch line on the driven member (Fig. 2). So, the sliding
motion on the driver tends to pull metal away from the pitch
line. On the other hand, on the gear the sliding tends to
compress the metal at the pitch line. Finally, the initial cracks
that form when a surface is severely loaded have a tendency to
intersect at the pitch line on the driver, while on the driven
member they do not. Secondly, the pinion, being smaller, has
obviously more cycles of operation than the gear. The slope of

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the fatigue curve makes the part with the most cycles the most
apt to fail [8].Basic parameters, influencing contact fatigue life
of spur gears can be summarized as Hertzian pressure on the
tooth flank and sliding conditions alongside the engagement
line.
Pressure on the tooth flank can be estimated with classical
Hertzian theory using an equivalent contact model (Fig. 3),
and can be expressed analytically by [6],
Where σH is Hertzian pressure, E is Young’s elastic modulus,
b is the gear width, ν is Poisson’s ratio, FN is the normal force
on the tooth flank and R* is the equivalent radius curvature
(Fig. 3).
Sliding alongside engagement line is due to a difference
between tangential components of velocity in the particular
gear contact points (Fig. 2a). As a rule, the maximum value of
tangential velocity difference occurs when the root of the teeth
(addendum) and the top of the counter-teeth (addendum) are
meshing. Wear is usually given as a numerical value
(parameter) and depends on the specific sliding coefficient ψ
(Fig. 2b).
However, additional, and also very important parameters are:
temperature on the tooth flank, thickness of the oil film in the
EHD lubrication regime, residual stresses in the area near the
top of the gear tooth flank surface, local geometry of the tooth
flank, roughness of the contacting profile etc.
Nevertheless, one of the main difficulties in numerical
modeling of contact fatigue at spur gears seems to be the
detailed determination of operational loading and/or strain
stress cycles of the meshing gears. Since the loading cycles are
very important for the determination of fatigue life and
directly influence the computational procedure of the initiation
damage definition, they must be precisely determined. For this
purpose, the following equivalent numerical contact model is
introduced.
2.2Equivelent numerical model
For computational determination of fatigue crack initiation at
gear teeth flanks, an equivalent contact model (Hertzian
theory) [10, 15, 17, 19, 24] is used (Fig. 3). The equivalent
cylinders have the same radii, as the curvature radii of gear
flanks at the observed point (the inner point of single teeth pair
engagement—point B).
According to Hertzian theory [19], the distribution of normal
contact pressure in the contact area can be determined by
where FN is the normal contact force per unit width, a is half
length of the contact area, which can be determined from [19]
where E* and R* are the equivalent Young’s elastic modulus
and the equivalent radius, respectively, defined as[19]
where E1, R1, ν1 and E2, R2, ν2 are the Young modulus, the
curvature radii and Poisson ratio of the contacting
cylinders(see Fig. 2).

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The maximum contact pressure p0=p(x=0) can then be
determined as [19]
In analyzing real mechanical components, some partial sliding
occurs during time dependent contact loading, which can
originate from different effects (complex loading conditions,
geometry, surface etc.) and it is often modeled by traction
force due to the pure Coulomb friction law [19]. In the
analyzed case, frictional contact loading q(x) is a result of the
traction force action (tangential loads)due to the relative
sliding of the contact bodies and is determined here by
utilizing the previously mentioned Coulomb friction law [19]
where μ is the coefficient of friction between contacting
bodies.
In regards to a general case of elastic contact between two
deformable bodies in a standstill situation, the analytical
solutions are well known. However, using general Hertzian
equations [19], it is difficult to provide the loading cycle
history and/or simulation of a contact pressure distribution of
moving contact in the analytical manner. Therefore, the finite
element method (FEM) issued for evaluating two-dimensional
friction contact with the aim of prescribing loading cycles at
the gears.
2.3 Fatigue crack initiation analysis
For accurate determination of the service life of gears,
loadings, which are in most cases random loadings of variable
amplitude, and the geometry of the gears and properties of
materials, of which the gears are made and which are not
known to be constants, have to be taken into account. The
more precise the modeling’s of these input parameters, the
more precise and reliable are the results.
In practice, the following two basic problems arise when
calculating the service life. Firstly, during designing, elements
and the entire products are optimized particularly with respect
to the service life. The basic requirement is that the service
lives of the individual elements are approximately equalized.
In this case, the model of occurrence of defects (crack
initiation) and the model of crack propagation are important
[2]. Secondly, a defect, i.e. an initial crack, is detected during
periodic inspection by a non-destructive method. If the
component concerned is not on stock or a considerable period
of time would be necessary to manufacture it, the interesting
fact is to know how long the damaged component will operate
with full rated loading and/or what the loading for the desired
service life is, i.e. the time necessary for the manufacture of a
new component gear.
When the stress loading cycles are determined, the fatigue
analysis for each observed material point can be performed.
The methods for fatigue analysis are most frequently based on
the relation between deformations, stresses and the number of
loading cycles and are usually modified to fit the nature of the
stress cycle, e.g. repeated or reversed stress cycle (Zahavi and
Torbilo [25]). The number of stress cycles required for a
fatigue crack to appear, can be determined iteratively with the
strain-life method ε-N, where the relationship between the
specific deformation increment, Δε, and the number of loading
cycles, Nf, is fully characterized with the following
equation[25]
where is the fatigue strength coefficient, b the strength
exponent, the fatigue ductility coefficient and c is the
fatigue ductility exponent. Generally, the following modified
approaches of the strain-life method are most often used for
fatigue calculations: Coffin-Manson’s hypothesis (ε-N
method), Morrow’s analysis, Smith-Watson-Topper (SWT)
method [25]. According to Morrow, the relationship between
strain amplitude, and pertinent number of load cycles to
failure, Nf, can be written as [21]
where E is the elastic modulus, is the fatigue strength
coefficient, is the fatigue ductility coefficient, b is

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exponent of strength and c is the fatigue ductility exponent,
respectively.
Morrow’s equation with mean stress, σm, correction [25]is
expressed as:
According to Coffin-Manson, this relationship can be
simplified as
Where σUTS is the ultimate tensile stress and D is the
ductility, defined as
The Smith-Watson-Topper method considers the influence of
the mean stress value and can be defined as [21]
where Δε1 is the amplitude of the maximum principal
deformation and is the maximum value of principal
stresses in the direction of maximum principal deformation.
All the described fatigue life models used in this work are
actually based on the strain-life method. Once a local stress or
strain-time history is established, a fatigue analysis method
has to be applied. Furthermore, material properties are
introduced as material fatigue data from tests [20]. The strain
life approach has an advantage over stress versus the number
of loading cycles (S-N) method, even in high cycle
application, due to its less scatter-prone materials data.
Assumptions that have been made in these fatigue life models
include the following: the models are uni-axial, which means
that just one principal stress occurs in one direction (single
stress vector). However, for more complex (multi axial)
loading conditions, multi axial critical plane analysis should
be applied.
Before damage can be determined and summed for each cycle,
a certain amount of correction needs to take place; the main
correction being the conversion of purely elastic stresses and
strains to elasto-plastic stresses and strains. In this work,
plasticity is accounted for in the crack initiation method by the
Neuber method [11, 25]. strain-life damage curve. Neuber’s
elastic-plastic correction is based on a simple principle that the
product of the elastic stress and strain should be equal to the
product of the elastic-plastic stress and strain from the cyclic
stress strain curve [11, 20]
where Δε, Δσ are incremental values of strain and stress, E is
the elastic modulus, Δεe is the incremental value of elastic
strain, n' is the material hardening exponent and K' is the
material strength coefficient. Using an iterative method, the
elastic-plastic stress and strain can be determined.

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Fig -4: Moving contact loading configurations in respect to
the initial crack position: a meshing of gear pair, b equivalent
contact model of meshing gears and c material model used for
determination of loading cycles
CONCLUSIONS
Gears are very specific parts of machines, and a lot of field
experience and testing has been done for over a century; the
standards for calculation, consequently, are more or less
detailed (AGMA, ISO, DIN). However, gears are subjected to
fatigue loads (surface contact fatigue and bending fatigue).
The kind of fatigue, which is the subject of the presented
work, is contact fatigue of gear teeth flanks. The paper
presents a general computational model for the determination
of initiation fatigue loading cycles in meshing gears. An
equivalent contact model of a cylinder and flat surface is used
for the simulation of contact fatigue crack initiation under
conditions of rolling and sliding contact loading of meshing
gears.
The material model is as summed to be homogenous, without
imperfections and/or inclusions ,and elastic shakedown is
considered. The modified Coffin-Manson method, in the
framework of finite element analysis (FEM), is used for
iterative analyses of contact fatigue crack initiation. The
number of loading cycles and places (on/under the contact
surface) for contact fatigue are presented.
Generally, regardless of the stress component selected, the
number of loading cycles required for initial fatigue damages
is in the range of Ni = 10-3 and where(on the contact surface
or subsurface) the contact fatigue damage first occurs mostly
depends on the coefficient of friction, material parameters and
contact geometry.
Although in final damage by pitting where small cracks form
in the tooth surface and then grow to the point where small,
round bits of metal break out of the tooth surface(Fig. 1), the
whole fatigue process should be estimated with the total
service life, i.e. Eq. (1).
Drawbacks of the proposed model are certainly then
considered residual stresses, elastic-hydrodynamic lubrication
conditions and material non-homogeneity. Also, some
additional experimental work should be provided for this
purpose. These are topics for further improvement of the
presented computational model.
ACKNOWLEDGEMENTS
Thanks to the expert who have contributed towards
development of the paper ―Computational approach to
contact fatigue damage initiation and deformation analysis of
gear teeth flanks‖. Would like to thanks prof. Bhargav
Makodia Venus International College of Technology
(Gandhinagar, Gujarat, India) and the faculty members who
contributed for the successful completion of work.
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1075 DOI 10.1007/s00170-005-0296-2
BIOGRAPHIES
The author is associated with Shri
Bhavshiji Polytechnique college,
Bhavnagar, Gujarat, India, as an assistant
professor.

Computational approach to contact fatigue damage initiation and deformation analysis of gear teeth fl anks

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Computational approach to contact fatigue damage initiation and deformation analysis of gear teeth fl anks